Solve 2x+3/x^2+x-6+x-1/x-2-3x-8/x+3

$$\frac { 2 x + 3 } { x ^ { 2 } + x - 6 } + \frac { x - 1 } { x - 2 } - \frac { 3 x - 8 } { x + 3 }$$

Evaluate

$$\frac{2\left(1-x\right)\left(x-8\right)}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-6.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{x-1}{x-2}-\frac{3x-8}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x-2 is \left(x-2\right)\left(x+3\right). Multiply \frac{x-1}{x-2} times \frac{x+3}{x+3}.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Since \frac{2x+3}{\left(x-2\right)\left(x+3\right)} and \frac{\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x+3+\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Do the multiplications in 2x+3+\left(x-1\right)\left(x+3\right).

$$\frac{2x+3+x^{2}+3x-x-3}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Combine like terms in 2x+3+x^{2}+3x-x-3.

$$\frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x+3 is \left(x-2\right)\left(x+3\right). Multiply \frac{3x-8}{x+3} times \frac{x-2}{x-2}.

$$\frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Since \frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)} and \frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x+x^{2}-\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in 4x+x^{2}-\left(3x-8\right)\left(x-2\right).

$$\frac{4x+x^{2}-3x^{2}+6x+8x-16}{\left(x-2\right)\left(x+3\right)}$$

Combine like terms in 4x+x^{2}-3x^{2}+6x+8x-16.

$$\frac{18x-2x^{2}-16}{\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+3\right).

$$\frac{18x-2x^{2}-16}{x^{2}+x-6}$$

Expand

$$-\frac{2\left(x^{2}-9x+8\right)}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-6.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{x-1}{x-2}-\frac{3x-8}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x-2 is \left(x-2\right)\left(x+3\right). Multiply \frac{x-1}{x-2} times \frac{x+3}{x+3}.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Since \frac{2x+3}{\left(x-2\right)\left(x+3\right)} and \frac{\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x+3+\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Do the multiplications in 2x+3+\left(x-1\right)\left(x+3\right).

$$\frac{2x+3+x^{2}+3x-x-3}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Combine like terms in 2x+3+x^{2}+3x-x-3.

$$\frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x+3 is \left(x-2\right)\left(x+3\right). Multiply \frac{3x-8}{x+3} times \frac{x-2}{x-2}.

$$\frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Since \frac{4x+x^{2}}{\left(x-2\right)\left(x+3\right)} and \frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x+x^{2}-\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in 4x+x^{2}-\left(3x-8\right)\left(x-2\right).

$$\frac{4x+x^{2}-3x^{2}+6x+8x-16}{\left(x-2\right)\left(x+3\right)}$$

Combine like terms in 4x+x^{2}-3x^{2}+6x+8x-16.

$$\frac{18x-2x^{2}-16}{\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+3\right).

$$\frac{18x-2x^{2}-16}{x^{2}+x-6}$$